Anisotropic bending-torsion coupling for warping in a non-linear beam

MEMS devices made from single crystal silicon often contain rod-like structures that are operated in bending and/or torsion. The design of these devices usually relies upon simple mechanical theories that ignore the coupling between these two modes of operation. In this paper, we develop a theory that is capable of accounting for the material coupling in the bending and twisting of single crystal beams which arises from anisotropic elastic properties and apply it in selected examples to the case of silicon. The generalized Saint-Venant torsion theory, which is valid for isotropic materials, is extended to arbitrary anisotropic linear elastic materials. The anisotropic material behavior couples the bending and torsion behavior. Thus, for the geometrically linear case, we find two warping functions associated with the bending moments and one warping function which is associated with the torsion moment. These warping patterns or functions are then taken as inputs to a geometrically non-linear formulation. Due to the presence of the additional warping functions, we find the existence of non-standard bi-moment and bi-shears which play an important role under certain conditions of extreme deformations. The final complexity of the non-linear formulation dictates the usage of a numerical solution procedure for practical computations. Here we employ a finite element scheme to solve the governing equations. Example computations elucidate the importance of the coupling effects by examining beams cut from (100) type silicon wafers.